A binomial probability experiment is conducted with the given parameters. Compute the probability of \( x \) successes in the \( n \) independent trials of the experiment. \( n=60, p=0.97, x=58 \) \( P(58)=\square \) (Do not round until the final answer. Then round to four decimal places as needed.)

To compute the probability of \( x \) successes in a binomial probability experiment, we can use the binomial probability formula: \[ P(x) = \binom{n}{x} p^x (1-p)^{n-x} \] Where: - \( \binom{n}{x} = \frac{n!}{x!(n-x)!} \) is the binomial coefficient, - \( p \) is the probability of success, - \( n \) is the number of trials, - \( x \) is the number of successes. Given: - \( n = 60 \) - \( p = 0.97 \) - \( x = 58 \) First, we compute the binomial coefficient \( \binom{60}{58} \): \[ \binom{60}{58} = \binom{60}{2} = \frac{60!}{2!(60-2)!} = \frac{60 \times 59}{2 \times 1} = 1770 \] Now, we can substitute into the formula: \[ P(58) = 1770 \times (0.97)^{58} \times (0.03)^{2} \] Calculating \( (0.97)^{58} \): \[ (0.97)^{58} \approx 0.2878 \text{ (using a calculator)} \] Calculating \( (0.03)^{2} \): \[ (0.03)^{2} = 0.0009 \] Now plug in these values: \[ P(58) = 1770 \times 0.2878 \times 0.0009 \] \[ P(58) \approx 1770 \times 0.00025902 \approx 0.458 \] Therefore, rounding to four decimal places gives: \[ P(58) \approx 0.4580 \] So, the probability \( P(58) \approx 0.4580 \).